Question

Given $\overrightarrow{A}=2\hat{i}+3\hat{j}$ and $\overrightarrow{B}=\hat{i}+\hat{j}$. The component of $\overrightarrow{A}$ along $\overrightarrow{B}$ is :

• A. $\frac{1}{\sqrt{2}}(\hat{i}+\hat{j})$
• B. $\frac{3}{\sqrt{2}}(\hat{i}+\hat{j})$
• C. $\frac{5}{\sqrt{2}}(\hat{i}+\hat{j})$
• D. $\frac{7}{\sqrt{2}}(\hat{i}+\hat{j})$

Answer 1

Answer: (C)

$\frac{5}{\sqrt{2}}(\hat{i}+\hat{j})$

$\overrightarrow{A}=2\hat{i}+3\hat{j}\overrightarrow{B}=\hat{i}+\hat{j}\angle \overrightarrow{A}\overrightarrow{B}=\theta \cos \theta =\frac{\overrightarrow{A}\cdot \overrightarrow{B}}{\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|}=\frac{2\times 1+3\times 1}{\sqrt{2^2+3^2}\sqrt{1^2+1^2}}=\frac{5}{\sqrt{13}\sqrt{2}}=\frac{5}{\sqrt{26}}\overrightarrow{A_B}=\left|\overrightarrow{A}\right|\cos \theta (\hat{i}+\hat{j})=\frac{\sqrt{13}\times 5}{\sqrt{26}}(\hat{i}+\hat{j})=\frac{5}{\sqrt{2}}(\hat{i}+\hat{j})$